Wire constructions of Abelian topological phases in three or more dimensions
Coupled-wire constructions have proven to be useful tools to characterize Abelian and non-Abelian topological states of matter in two spatial dimensions. In many cases, their success has been complemented by the vast arsenal of other theoretical tools available to study such systems. In three dimensions, however, much less is known about topological phases. Since the theoretical arsenal in this case is smaller, it stands to reason that wire constructions, which are based on one-dimensional physics, could play a useful role in developing a greater microscopic understanding of three-dimensional topological phases. In this paper, we provide a comprehensive strategy, based on the geometric arrangement of commuting projectors in the toric code, to generate and characterize coupled-wire realizations of strongly-interacting three-dimensional topological phases. We show how this method can be used to construct pointlike and linelike excitations, and to determine the topological degeneracy. We also point out how, with minor modifications, the machinery already developed in two dimensions can be naturally applied to study the surface states of these systems, a fact that has implications for the study of surface topological order. Finally, we show that the strategy developed for the construction of three-dimensional topological phases generalizes readily to arbitrary dimensions, vastly expanding the existing landscape of coupled-wire theories. Throughout the paper, we discuss topological order in three and four dimensions as a concrete example of this approach, but the approach itself is not limited to this type of topological order.
- I Introduction
II Three-dimensional wire constructions
- II.1 Decoupled wires
- II.2 Interwire couplings and criteria for producing gapped states of matter
- II.3.1 Change of basis
- II.3.2 Compactification, vertex operators, and fractional charges
- II.3.3 Pointlike and linelike excitations
- II.3.4 Energetics of pointlike and linelike defects
- II.3.5 Statistics of pointlike and linelike defects
- II.3.6 Topological ground-state degeneracy
- II.3.7 Topological field theory
- II.4 Example: topological order in three-dimensional space from coupled wires
- III Higher-dimensional wire constructions
- IV Conclusion
- A Deconfinement of defects along the direction of a wire
- B Discrete gauge symmetry and ground state in the limit of vanishing kinetic energy
The experimental discovery of the integer and fractional quantum Hall effects excited enormous interest in the study of topological states of matter in two dimensional space. Strongly interacting states of matter distinguished by the presence of excitations with fractional quantum numbers or nontrivial boundary modes have attracted particular attention from theorists. Over time, a vast arsenal of theoretical tools has been developed to study such systems, from the microscopic (e.g., numerical techniques to study lattice models with topologically ordered ground states) to the macroscopic (e.g., topological quantum field theories).
Wire constructions, which were first undertaken for the integer Poilblanc et al. (1987); Yakovenko (1991); Lee (1994), and later the fractional Kane et al. (2002); Teo and Kane (2014); Mong et al. (2014); Klinovaja and Loss (2014); Meng et al. (2014); Klinovaja and Tserkovnyak (2014); Sagi and Oreg (2014); Neupert et al. (2014); Meng and Sela (2014); Klinovaja et al. (2015); Santos et al. (2015); Sagi et al. (2015), quantum Hall effect, are conveniently poised midway between these two extremes. The approach in this case is to model a topological phase by starting from an anisotropic theory of decoupled gapless quantum wires, and then introducing local couplings between the wires to produce a gapped state of matter with an isotropic low-energy description. This approach has the virtue of yielding the edge theory, itself that of a Luttinger liquid, directly, and of providing means to construct the low-lying quasiparticle excitations of the bulk quantum liquid. Furthermore, because wire constructions make use of well-understood techniques in one-dimensional physics, such as Abelian (or non-Abelian) bosonization, one can construct analytically tractable theories of states of matter that might not otherwise admit a controlled analytical description.
In recent years, wire constructions have also been used to study fractional topological insulators (FTIs) Neupert et al. (2014); Sagi and Oreg (2014); Santos et al. (2015) and spin liquids Meng et al. (2015); Gorohovsky et al. (2015), and also to develop an extension Neupert et al. (2014) of the ten-fold way for noninteracting fermions Altland and Zirnbauer (1997); Schnyder et al. (2009); Kitaev (2009); Ryu et al. (2010) to strongly-correlated systems.
Since the prediction Fu et al. (2007) and discovery Hsi (); Hsieh et al. (2009); Che (a) of three-dimensional topological insulators (TIs), there has been a growing interest in understanding topological states of matter in three spatial dimensions. In addition to generalizing these time-reversal invariant topological insulator to the strongly-interacting regime Maciejko et al. (2010); Swingle et al. (2011); Maciejko et al. (2014), there has been an effort to derive effective field theories describing the bulk of such TIs, and to determine the bulk-boundary correspondence in such theories that yields the hallmark single Dirac cone on the two-dimensional surface Cho and Moore (2011); Chan et al. (2013); Tiwari et al. (2014); Ye and Gu (2015); Che (b). Further work has undertaken efforts to understand broader features of three-dimensional topological states of matter, such as the statistics of pointlike and linelike excitations Wang and Levin (2014); Lin and Levin (2015). For example, it has been shown that certain three-dimensional topological phases can only be distinguished by the mutual statistics among three linelike excitations Lin and Levin (2015).
Another major direction of work concerns three-dimensional systems whose surfaces are themselves two-dimensional topological states of matter. The simplest example of this phenomenon occurs on the surface of a TI when time-reversal symmetry is locally broken by a magnetic field on the surface, in which case a half-integer surface quantum Hall effect develops Fu and Kane (2007); Qi et al. (2008); Xu et al. (2014); Yoshimi et al. (2015). Further theoretical work has shown that generic three-dimensional topological phases, including but not limited to the fermionic TI, can exhibit more exotic surface topological phases that cannot exist with the same realization of symmetries for local Hamiltonians in purely two-dimensional space. This family of surface phenomena is known as surface topological order von Keyserlingk et al. (2013); Vishwanath and Senthil (2013); Wang and Senthil (2013); Wang et al. (2013); Burnell et al. (2014); Chen et al. (2014); Metlitski et al. (2015); Mross et al. (2015). Several recent works Mross et al. (2015); Sahoo et al. (2015) have approached the question of surface topological order by applying the quasi-one-dimensional physics of wire constructions, although it appears that this approach necessitates the use of an unusual “antiferromagnetic” time-reversal symmetry rather than the usual (physical) realization of reversal of time, which acts on-site. It is possible that a fully three-dimensional wire construction could remedy this peculiarity, although such a description is still lacking.
Layer constructions, in which planes of two-dimensional topological liquids are stacked on top of one another and coupled, were used to construct the single surface Dirac cone of the three-dimensional TI Hosur et al. (2010) and to study surface topological order Jian and Qi (2014). Wire constructions of three-dimensional topological states of matter have also recently been undertaken, yielding Weyl semimetals Vazifeh (2013); Meng (2015) and a class of fractional topological insulators Sagi and Oreg (2015). However, in all three cases, different methods are used to develop the wire constructions themselves, and little effort has been made to extend these constructions beyond the specific problem at hand in each example. In order to attack the most distinctive aspects of topological states of matter in three dimensions, such as surface topological order, it is therefore necessary to develop a framework that lends itself readily to a variety of approaches with minimal modifications.
In this paper, we provide a comprehensive strategy to design wire constructions of strongly-interacting Abelian topological states of matter in three dimensions. The strategy that we present is to start with decoupled quantum wires placed on the links of a two-dimensional square lattice, and then to couple the wires with many-body interactions associated with each star and plaquette of the lattice. In this way, each interaction term that couples neighboring wires can be viewed as corresponding to one of the commuting projectors that enters Kitaev’s toric code Hamiltonian Kitaev (2003). This correspondence simplifies the application of a criterion, first proposed by Haldane, to ensure that these interaction terms do not compete, and are sufficient in number to gap out all gapless modes in the array of quantum wires when periodic boundary conditions are imposed along all three spatial directions.
When all interaction terms satisfy this criterion, the Hamiltonian is frustration-free, and taking the strong-coupling limit produces a gapped three-dimensional state of matter. With this done, one can proceed to characterize this state of matter in terms of its pointlike and linelike excitations, as well as their statistics, and calculate the topological degeneracy, if any, of the ground-state manifold. The class of three-dimensional models studied in this work features a topological degeneracy given by , where the integer-valued matrix contains information about the mutual statistics of pointlike and linelike excitations in the theory. This is in close analogy with the -matrix formalism developed for two-dimensional topological states of matter Wen and Zee (1992). When periodic boundary conditions are relaxed by the presence of two-dimensional terminating surfaces, we further show that gapless surface states result. One can apply the coupled-wire techniques already developed in two dimensions to study the various gapped surface states that can be produced by introducing interwire hoppings or interactions on the surface, provided that the added terms are compatible with the interactions in the bulk.
In addition, we show that the above strategy for constructing three-dimensional Abelian topological states of matter can be readily extended to arbitrary dimensions, vastly expanding the existing scope of the coupled-wire approach. Indeed, much as it is possible to define higher-dimensional versions of the toric code on hypercubic lattices (see, e.g., Ref. Mazáč and Hamma (2012)), one can arrange a set of decoupled quantum wires on a -dimensional hypercubic lattice and couple them with interactions defined on stars and plaquettes of this lattice. Applying Haldane’s compatibility criterion, one can show that these interactions produce a gapped -dimensional state of matter, whose excitations and topological properties can be investigated much as in the three-dimensional case.
The structure of this paper is as follows. In Sec. II, we develop in detail the strategy discussed above for constructing three-dimensional topological phases from coupled wires. In Sec. II.1, we establish the basic notation used to describe the array of decoupled quantum wires. In Sec. II.2, we present Haldane’s compatibility criterion and a class of many-body interactions between wires that satisfy it. (This class is mainly chosen for analytical expedience, and is not the only class of interactions that can be constructed according to our strategy.) In Sec. II.3, we show how to use the interacting arrays of quantum wires defined in Secs. II.1 and II.2 to study states of matter with fractionalized excitations. In particular, we show how to construct pointlike and linelike excitations, and determine their statistics, as well as the topological ground state degeneracy. Next, in Sec. II.4 we exemplify our strategy with perhaps the simplest type of topological order in three dimensions, namely topological order. Furthermore, we investigate the surface states of these -topologically-ordered states of matter, and find that they are unstable to interwire hoppings. Additionally, a surface fractional quantum Hall effect with Hall conductivity can develop at the expense of breaking time-reversal symmetry on the surface. We also discuss how these observations regarding surface states can be extended to the more general class of interwire interactions introduced in Sec. II.2.
Next, in Sec. III, we outline the generalization of our results to arbitrary dimensions. In Sec. III.1, we discuss how to define -dimensional hypercubic arrays of quantum wires that are analogous to the square array of quantum wires used to construct three-dimensional topological states. Then, in Sec. III.2, we generalize the results of Sec. II.2 regarding the definitions of appropriate interwire couplings and their compatibility in the strong-coupling limit. Finally, in Sec. III.3, we provide an example of this generalization by constructing -topologically-ordered states of matter in four dimensions, and constructing their pointlike, linelike, and membranelike excitations, before concluding in Sec. IV.
Ii Three-dimensional wire constructions
In this section, a method to construct arrays of coupled wires realizing topological phases of matter in three-dimensional space is presented. We begin by defining a class of gapless theories describing decoupled wires, before moving on to a discussion of interwire couplings. In particular, we provide a set of algebraic criteria that are sufficient to determine whether the theory is gapped when periodic boundary conditions are imposed.
ii.1 Decoupled wires
We consider a two-dimensional array of quantum wires, labeled by Latin indices , placed on the links of a two-dimensional square lattice embedded in three-dimensional Euclidean space. Each quantum wire is assumed to be gapless and nonchiral, and therefore to contain gapless degrees of freedom, labeled by Greek indices . We take the wires (of length ) to lie along the -direction, and the square lattice to lie in the - plane. We will impose periodic boundary conditions in all directions (, , and ) until further notice. The set of decoupled quantum wires is described by the quadratic Lagrangian
|is a vector that collects the scalar fields defined in each of the wires. We use vertical bars as a visual aid to separate degrees of freedom defined in different wires. The block-diagonal -dimensional matrix|
|where is the unit matrix of dimension and is a symmetric matrix with integer entries, yields the equal-time commutation relations|
|We will omit the explicit time dependence of the fields from now on. Finally, the block-diagonal matrix|
|where the matrix is real, symmetric, and positive-definite. The matrix is set by microscopics within each wire, and will usually be taken to be a diagonal matrix in this work. However, the matrix , which enters the commutation relations (1d), contains crucial data that define the fundamental degrees of freedom in a wire. The final data necessary to complete the definition of the theory describing the two-dimensional array of decoupled quantum wires is the -dimensional “charge-vector”|
The -dimensional integer vector collects the electric charges associated with the scalar fields , .
The theory defined by Eqs. (1) can be viewed as an effective low-energy description of a two-dimensional array of decoupled physical quantum wires containing fermionic or bosonic degrees of freedom.
For fermions, each wire contains flavors of chiral scalar fields and , where label right- and left-moving degrees of freedom, respectively. These fields obey the chiral equal-time commutation relations
|and therefore, for fermions, the matrix entering Eq. (1d) is given by|
|We further adopt the convention that the charge-vector|
in units where the electron charge is set to unity, for a fermionic wire with channels. Treating an array of fermionic quantum wires within Abelian bosonization, as we do here, further requires the use of Klein factors, which are needed in order to assure that fermionic vertex operators (defined below) defined in different wires anticommute with one another. These Klein factors can be subsumed into the equal-time commutation relations for the scalar fields and . This can be done by integrating both sides of Eqs. (2a) over all and fixing the arbitrary constant of integration to be the Klein factor necessary to ensure the appropriate anticommutation of vertex operators. We refer the reader to the Appendix of Ref. Neupert et al. (2011) for more details on this procedure.
For bosons, each wire instead contains flavors of nonchiral scalar fields and , where label “charge” and “spin” degrees of freedom, respectively. These fields obey the equal-time commutation relations
|so that the -matrix for bosons is|
|We take the bosonic charge vector to be|
in units where the electron charge is set to unity, so that the fields carry a electric charge, while is neutral. (Of course, one could define a “spin vector” analogous to that encodes the coupling to another gauge field for spin, but, for simplicity, we will work exclusively with electric charges here.)
The fundamental excitations of a fermionic or bosonic wire can be built out of the vertex operators
for any and , where we have adopted the convention of summing over repeated indices. Any local operator acting within a single wire can be built from these vertex operators. Similarly, operators spanning multiple wires can be built by taking products of vertex operators from each constituent wire. The charges of the excitations created by these vertex operators are measured by the charge operator
for any and , where is the length of a wire. The normalization of the charge operator is taken to be such that
at equal times, indicating that the vertex operator carries the charge .
ii.2 Interwire couplings and criteria for producing gapped states of matter
Given the two-dimensional array of decoupled and gapless quantum wires defined in Sec. II.1, we would like to devise a systematic way of introducing strong single-particle or many-body couplings between adjacent wires in order to yield a variety of gapped topologically-nontrivial three-dimensional phases of matter. Our strategy will be to extend the approach taken in Ref. Neupert et al. (2014), which considered one-dimensional chains of wires, to two dimensions. We begin by adding to the quadratic Lagrangian defined in Eq. (1) a set of cosine potentials
Here, the -dimensional integer vectors encode tunneling processes between adjacent wires. This interpretation becomes transparent upon recognizing that, up to an overall phase,
where are the vertex operators defined in Eq. (4). [We follow Ref. Neupert et al. (2014) in using the shorthand notation and in employing an appropriate point-splitting prescription when multiplying fermionic operators.] For generic tunneling vectors , Eq. (8) describes a many-body or correlated tunneling that amounts to an interaction term in the Lagrangian
The real-valued functions and in Eq. (7) encode the effects of disorder on the amplitude and phase of these interwire couplings.
Distinct states of matter can be realized by restricting the sum over tunneling vectors in Eq. (7) to ensure that the interaction terms (8) satisfy certain symmetries. For all examples considered in this work, we will assume that either charge or number-parity conservation holds. The former is imposed by demanding that
|while the latter is imposed by relaxing the above requirement to|
For a detailed discussion of how further symmetry requirements constrain the tunneling vectors , see Ref. Neupert et al. (2014).
We are now prepared to discuss the strategy we employ to produce gapped states of matter from the above construction. We first recall that the array of decoupled quantum wires consists of gapless degrees of freedom. As noted in Ref. Haldane (1995), and later employed in Refs. Neupert et al. (2011, 2014), a single cosine term in the sum in Eq. (7) is capable of removing (i.e., gapping out) at most two of these gapless degrees of freedom from the low-energy sector of the theory. This occurs in the limit , where the argument of the cosine term becomes pinned to its classical minimum. Therefore, in principle it takes only cosine terms to gap out all degrees of freedom in the bulk of the array of quantum wires when periodic boundary conditions are imposed. Matters are complicated somewhat by the nontrivial commutation relations (1d), which ensure that cosine terms corresponding to distinct tunneling vectors and do not commute in general. Consequently, it is possible that quantum fluctuations may lead to competition between the various cosine terms that frustrates the optimization problem of simultaneously minimizing all of these terms. However, in Ref. Haldane (1995), Haldane observed that if the criterion
holds, then the cosine terms associated with the tunneling vectors and can be minimized independently, and therefore do not compete with one another. [Note that each tunneling vector must also satisfy Eq. (11), i.e., we require that for all .] Therefore, if one can find a “Haldane set” of linearly-independent tunneling vectors, all of which satisfy Eq. (11), then it is possible to gap out all degrees of freedom in the array of quantum wires by adding sufficiently strong interactions of the form (7). If such a set is found, then it suffices to restrict the sum in Eq. (7) to , and to posit that all couplings are sufficiently large in magnitude to gap out all modes in the array of quantum wires.
We now present a simple geometric prescription to aid in the determination of the existence (or lack thereof) of a Haldane set for a two-dimensional array of quantum wires with some set of desired symmetries. This prescription capitalizes on the fact that we have chosen all quantum wires to lie on the links of a square lattice. (In principle, this is not the only possible choice of lattice geometry, but it provides a simple way of counting degrees of freedom in any dimension, as we will see below and in Sec. III.) On a square lattice with sites, there are “stars” (centered on the vertices of the lattice) and “plaquettes” (centered on the vertices of the dual lattice), assuming that periodic boundary conditions are imposed as in Fig. 1. If we associate the tunneling vectors and with each star and plaquette , respectively, then we have a set of tunneling vectors. Since there are gapless degrees of freedom in the array of decoupled quantum wires, we can obtain the necessary number of tunneling vectors by expanding this set to include “flavors” of tunneling vectors and for each star and plaquette, respectively. We label these flavors using a teletype index . Imposing the Haldane criterion (11) on this set of tunneling vectors then yields the set of equations
If the above equations are satisfied, then the set of tunneling vectors is a Haldane set, and therefore capable of yielding a gapped phase in the strong-coupling limit.
We now turn to the problem of building tunneling vectors and . Enumerating all solutions to this problem for all matrices is beyond the scope of the present work. However, we will present below one way of constructing these tunneling vectors that builds in the minimal symmetries of charge and/or parity conservation [Eqs. (10)] and greatly reduces the number of equations that must be solved [relative to Eqs. (12), which contain an infinite number of linear equations in the thermodynamic limit if no additional information is provided]. In particular, if we desire charge conservation [Eq. (10a)] to hold, we may define the tunneling vectors by their nonvanishing components
where we recall that labels the quantum wires and labels the degrees of freedom within a wire. Here, , , , and are arbitrary -dimensional integer vectors. The Kronecker deltas in the tunneling vector ensure that its nonzero entries are defined within the quantum wires to the north, , west of the vertex on which star is centered. The Kronecker deltas in select the quantum wires , which are defined similarly for the plaquette (see Fig. 1). With these definitions, one verifies that Eq. (10a) holds independently of the form of , , and the charge-vector for a single wire.
Similarly, when we wish to impose number-parity conservation [Eq. (10b)], we may define for any and any
and verify that Eq. (10b) holds independently of the form of , , and .
Henceforth, we will focus on the charge-conserving tunneling vectors defined in Eqs. (13), as all general criteria discussed below have analogues for the parity-conserving tunneling vectors defined in Eqs. (14).
The charge-conserving tunneling vectors defined in Eqs. (13) are expressed in a convenient pictorial form in Fig. 2. From this pictorial representation, it is clear that any two distinct, adjacent stars (be they of the same flavor or different flavors) share a single wire between them. The same statement holds for plaquettes. However, adjacent stars and plaquettes share two wires between them, regardless of the flavor. Therefore, one can show that Eqs. (12) are satisfied if and only if
for all j and and . Equations (15) are fundamental to our construction, as each solution to these equations for a given dimension of the matrix may in principle describe a distinct gapped phase of matter.
Observe that Eqs. (15) are symmetric under and . Therefore, these criteria amount to a set of linear equations in variables. This is important for two reasons. First, the number of equations does not scale with the number of quantum wires in the array. This ensures that a single solution to these equations holds for any system size when periodic boundary conditions are imposed. Second, this set of equations is underconstrained for any (i.e., there are always more variables than equations). This means that for generic matrices of fixed dimension , there is in principle more than one solution to Eqs. (15).
We aim to construct gapped states of matter that have an isotropic low-energy description. Consequently, it is natural to demand that the tunneling vectors defined in Eqs. (13) and depicted in Fig. 2 are independent of direction. This can be achieved by imposing the additional constraints
Note that Eq. (15c) is solved independently of the form of the -dimensional vectors and if Eqs. (16) hold. These constraints reduce the total number of variables contained in the tunneling vectors and from to , and the number of nontrivial equations to , i.e.,
which are merely rewritings of Eqs. (15a) and (15b). With this, we have arrived at the simplest incarnation of our construction. We will henceforth assume that Eqs. (17) hold for appropriate choices of the , -dimensional vectors and . However, note that Eqs. (16) are sufficient but not necessary in order to produce a state of matter that has an isotropic low-energy description. We will therefore comment, as appropriate, on how our results below generalize to cases where and .
ii.3.1 Change of basis
In this section, we outline how to use two-dimensional arrays of coupled quantum wires, like those described in the previous two sections, to study phases of matter with fractionalized excitations. To this end, let us assume that we have a Haldane set containing tunneling vectors and with defined by Eqs. (13) that satisfy (16) and the Haldane criterion (17). With these assumptions, the two-dimensional array of coupled quantum wires acquires a gap in the strong-coupling limit, yielding a three-dimensional gapped state of matter.
As discussed in the previous section, the phase of matter obtained in this way is a system of strongly-interacting fermions or bosons. However, for the purposes of studying fractionalization, it is convenient to work in a basis where the “fundamental” constituents of each wire are not fermions or bosons, but (possibly fractionalized) quasiparticles. This is achieved by making the change of basis
|for some invertible integer-valued matrix .|
This change of variables has several virtues. First, remains symmetric and integer valued. Second, remains integer valued. Third, this change of variables leaves the quantity , which enters the argument of the cosine terms in Eq. (7), invariant, i.e.,
Thus, the linear transformation (18) does not change the character of the interaction itself, although it alters the tunneling vector and the -dimensional vector of bosonic fields. Furthermore, one verifies that the linear transformation (18) does not alter the compatibility criteria (15) or the quantity that determines the presence or absence of charge or number-parity conservation.
Given the possibility of performing a change of basis of the form (18), we may now take a different approach. Instead of viewing the wire construction as a theory, with the Lagrangian (9), of scalar fields obeying the commutation relations (1d) with a -matrix [Eq. (2)] for fermions or [Eq. (3)] for bosons, we may also view it as a theory, with the Lagrangian
of scalar fields obeying the new equal-time commutation relations
for and , which are neither fermionic nor bosonic in nature. We allow to be any symmetric, invertible, integer matrix, as long as it is related to or by a transformation of the form (18). Interactions between wires that yield a gapped state of matter can be constructed by following the procedures of the previous section. The integer tunneling vectors and obtained in this way form a Haldane set related to the Haldane set by the transformation (18). For reasons of simplicity that will become clear momentarily, we will concern ourselves in this paper primarily with the tunneling vectors and whose nonzero entries are equal to . (Of course, nothing prevents us from also considering cases where this does not hold.) The counterparts and of these tunneling vectors under the transformation (18) generically have entries with magnitude larger than 1. This fact will be of importance to us now, as we turn to the issue of compactification.
ii.3.2 Compactification, vertex operators, and fractional charges
Although the transformation (18) might appear innocuous, there is a fundamental difference between the theory with the Lagrangian defined in Eq. (20) and the original fermionic or bosonic theory with the Lagrangian defined in Eq. (9) when periodic boundary conditions are imposed in the -direction (as we have assumed from the outset). In the latter theory, which is a theory of interacting electrons or bosons treated within bosonization, the traditional choice of compactification for the scalar fields with and is
for . This choice ensures the single-valuedness of the fermionic or bosonic vertex operators (4) under , and, in turn, that of the Lagrangian , as one can re-write in terms of the correlated tunnelings (8), which reduce to products of these vertex operators. However, depending on the tunneling vectors , there may be other, less stringent, compactifications of these scalar fields that also render the Lagrangian single-valued under . The parsimonious course of action is to choose the “minimal” compactification, i.e., the smallest compactification radius that still maintains the single-valuedness of under .
If the tunneling vectors and correspond to the tunneling vectors and under the transformation (18) whose only nonzero entries are equal to , then there is a clear choice of minimal compactification. This choice can be obtained as follows. Working in the tilde basis, we can rewrite the interactions using the relation [analogous to (8)]
|thereby implicitly defining a new set of fermionic or bosonic vertex operators,|
The minimal compactification is then obtained by demanding that this new set of vertex operators be single valued under . For any and , this is achieved by imposing the periodic boundary conditions
for . Here, there is an important difference with respect to Eq. (22). Because is an integer-valued matrix, is generically a rational-valued matrix. The field is thus allowed to advance by rational (rather than integer) multiples of when the coordinate is advanced through a full period . This crucial distinction is what allows for the existence of fractionally-charged operators in the coupled wire array, as we now demonstrate.
Fractional quantum numbers appear in the wire construction because the compactification condition (24) allows for the existence of “quasiparticle” vertex operators
for any and that are multivalued under the operation . The fact that these vertex operators generically carry fractional charges can be seen by considering the transformed charge operator
for any and . Its normalization is here chosen such that the fermionic or bosonic vertex operators defined in Eq. (23b) have charge . Indeed, for any and , the equal-time commutator
indicates that, since is generically a rational matrix, the quasiparticle operator generically has a rational charge. In particular, if under the transformation (18), and has at least one rational entry with magnitude smaller than 1, the operator must then carry a fractional charge.
ii.3.3 Pointlike and linelike excitations
We now outline the relationship between the quasiparticle vertex operators defined in Eq. (25) and (possibly fractionalized) excitations in the array of coupled quantum wires. In the strong-coupling limit , the compatibility criteria (15) ensure that the quantity
where or , is pinned to a classical minimum of the corresponding cosine potential in . Following Refs. Kane et al. (2002); Teo and Kane (2014) and subsequent works, we identify excitations in the coupled-wire theory with solitons that increment the “pinned field” by an integer multiple of . These excitations can therefore be viewed as living on either the stars or the plaquettes of the square lattice, rather than within the wires themselves.
We now demonstrate that products of an appropriate number of quasiparticle vertex operators of the form (25) can be used to move the soliton defects to adjacent stars and plaquettes. To see this, we write out the pinned fields explicitly for all tunneling vectors with defined on the stars ,
|and for all tunneling vectors with defined on the plaquettes ,|
For any star or plaquette from the square lattice and for any , observe that, by Eq. (21), the equal-time commutators
hold. Here, the uppercase Latin index labels the four cardinal directions. Equation (30) indicates that the pair of fields and can be viewed, up to a multiplicative constant, as canonical conjugates to the pair of fields and that enter the pair of pinned fields and , respectively. Interpreted this way, Eqs. (30) suggest that, for any , , and , the operators
act on the pinned fields as [ denotes the magnitude squared of the vector ]
and [ denotes the magnitude squared of the vector ]
where is the Heaviside step function. If the underlying quantum wires in the theory are fermionic, the arbitrary integration constants above are identified with Klein factors that are necessary in order to ensure the anticommutation of fermionic vertex operators in different wires. If the underlying wires are bosonic, however, the integration constants can be set to zero.
Evidently, the operators and defined in Eqs. (31) create solitons in the pinned fields and , respectively. However, the link on star is shared with the star adjacent to along the cardinal direction , and, likewise, the link on plaquette is shared with the plaquette adjacent to along the cardinal direction . Therefore,
respectively. Note the sign difference with respect to Eqs. (32) and (33). This difference also stems from Fig. 2. Consequently, we interpret the operators and as creating a soliton-antisoliton pair straddling the links and , respectively (see Fig. 3). By taking the derivative with respect to of Eqs. (32), (33), (35), and (36), we can interpret the operators and as creating a dipole in the soliton density across the links and , respectively. Correspondingly, the annihilation vertex operators and reverse the orientations of these dipoles in the soliton density.